I want to display a list from another site on our sharepoint server. The Page Viewer Web Part looks like my only option and it shows - BUT, it shows the entire web page, not just the list (in the specific view I want).

The OOTB sharepoint list viewer web part can only show lists within same site. So you need to create your own custom web part to show lists from another site. Also with some effort you can use content query webpart with JQuery/Get to request the page and hide all the other contents in the page and show just the list. There is a commercial tool which does that purpose here (and I don't work for them) - http://www.kwizcom.com/sharepoint-add-ons/sharepoint-remote-list-viewer/overview/NLV - MCTS - Blog - Twitter - In